Welcome to the board.
I've done round trips from CHI to NM (twice), and from MN to FL a couple of times, so i'll offer a few suggestions. first, get the sleeper if you can. you can see if on-board upgrades are available when you get to NYC. For me, getting in the sleeper means the difference between the increased potential for the Endurance contest that is coach ( air coach, bus coach, train coach - doen't matter) travel in all its forms, and a reasonable expectation of a decent trip.
2. "Will time drag by, or go quickly?" Depends upon your attitude. if you're one of those folks who just can't get where they're going fast enough, and if you view travel as simply a means from A to B, then time may drag a bit. Kind of like what happens on long drives, if you've ever driven someplace overnight. If you look at your train trip as an integral part of the overall trip, then things will be more pleasant. Travelling by train also enables one to bring along items that will keep the youger ones occupied. monopoly is good, as are decks of cards. The stops tend to go by quickly. In any case, another item to bring with you would be those "wet-naps" - the pre-moistened towelettes that folks with toddlers bring along for cleaning up all the messes. It feels good to freashen up a bit after a while on the trian, and, if you're in coach, you won't have access to the shower facilities (think: sponge bath if you want to wash up while in coach. Just take care to lock the door while you're using the washroom).
3. Books and other reading material are good to pass the time.
4. By all means, you can get up and move about the train while it is in motion! This is one of the advantages that train travel has over air travel - you can go to the lounge car to socialize or look out the window to see what scenery there is to see, or to have a libation or two (CAUTION: Don't over-indulge. The crew won't tolerate that sort of behaviour). The only difference is that sleeping car passengers have access to the sleeper section; coach passengers are denied access to the sleepers. Everyone has access to the diner
5. The train get heavy use throughout the year, and I'm sorry to say that some cars show this heavy use. Most of what I'm talking about here will be "cosmetic" issues, such as well-worn carpeting. With several exceptions, the cars are in for maintenance only every 180 days or so. The rest of the time they are out on the road, getting as much as 250,000 miles per year of travel. In all fairness, Amtrak has made many improvements the past decade or so, and what you may have heard from the neighbor who took a trip last year, or years ago, will be out-of-date information. I've travelled on airplanes that haven't seen the vacuum cleaner for several weeks, and many people's private cars could use some serious cleaning as well. As for the bathrooms, it depends. In coach, there is only one crewmember for every three cars, and they don't have time to clean the bathrooms after every single passenger. While the train crew usually makes their best efforts to keep things neat and tidy, there are passengers who are slobs, or who act as though they expect the butler to clean up after them. There is no butler service aboard the train, so clean up after yourself. Most bathrooms will be clean, but with a busy train there always seem to be one or two rapscallions onboard who ruin it for everyone else.
6. Like every other mode of travel, trains have their good points and their bad ones. Many of the critisms (tardiness, rude employees, slovenliness, for example) that are levellled against the passenger train service provider could very, very easiy be made regarding airplanes, and certain airlines, and car travel as well. Some folks get aboard expecting every train to be just like the Orient Express of years gone by. The truth is that even during the Golden Age of Trains, most trains didn't have that level of luxurious extravagance then, and don't now. Even back in years gone by, that level of opulence was unavilable at common-carrier prices. if you're looking for the Orient Express, you should book travle over the AOE (if they're still in business), and not Amtrak. Amtrak travels over privately owned track, and sometimes the private railroads (who are the landlords in this case) will give priority to their own freight trains over Amtrak. Also, don't board with the expectation of the plastic smile that we all see from the airlines. With Amtrak, you get to see the crew as real human beings. Some train crew are fantastic, some are adequate, and some train crew should clearly be working somewhere else.
You may also see first-hand how the private railroads have to pay for traffic control and road maintenance entirely by themselves - unlike the highway system, which is paid for using the gas tax and a well-endowed gov't-supported Highway Trust Fund, or the airlines, which have their traffic control and airports paid for by the FAA and its well-endowed gov't supported Trust Fund, or the Municipal Transit Authority. There is no trust Fund for passenger rail, no dedicated source of funding for Amtrak. Sometimes this translates into your train being delayede because another train is occupying the single stretch of track ahead of you, and the entire line is choking with freight traffic. Just be patient, Amtrak will be working to get you through to your destination. You see, just as with highways and airports, there can be traffic jams on the RR. The main difference here is that the RR has to buy and maintain all of their own track, they (usually) don't have the option of dipping into Federal funding to get additional capacity built.
I don't want to discourage you from taking the train, though I understand that this post could be misconstrued as such. i hope that you're expereinces will be as positive as mine have been (with a couple of exceptions that were the fault of the landlord RR, not Amtrak). Just remember to have a Positive Mental Attitude, and to try to roll with the puches, and tdon't swear off train travel just because you had one lousy trip ( as I mentioned before, it wasn't the fault of Amtrak. It was due to the landlord RR making us take the sidetrack all of the time).
We look forward to learning of yur experiences when you return. Oh, yeah, try to upgrade to the sleeper for the overnight portion of your trip. Even the economy sleeper will feel better than sleeping in coach, and ANYTHING will feel superior to airline coach. Sleeping supine feels better than sleeping in a recline position.